On the other hand, Hilfer [1]} introduced
a fractional derivative \(\mathcal {D}_{a+}^{\eta ,\nu }(\cdot )\)  having two parameters \(\eta \in (n-1, n), n \in \mathbb {N}\)  and \(\nu ~(0\le \nu \le 1)\)  which in specific gives the Riemann–Liouville and the Caputo derivative operator.
Furati et al. [2]} analyzed nonlinear FDEs with Hilfer derivative operator. Sousa and Oliveira [3]}, [4]} introduced a new definition of fractional derivative viz. \(\Psi \) –Hilfer fractional derivative and investigated its important properties. Further, it is proved that the \(\Psi \) –Hilfer derivative is the generalization of many existing fractional derivative operators. In [4]}, Sousa and Oliveira established generalized Gronwall inequality through the fractional integral with respect to another function and studied the existence, uniqueness and data dependence of solution of Cauchy problem with \(\Psi \) –Hilfer fractional derivative. For the study relating to existence, uniqueness and stability of different sorts of FDEs involving \(\psi \) -Hilfer fractional derivative operator, we refer to the work of Sousa et al.[6]}, [7]}, [8]} and the references given there in.
